Semantics For Disjunctive Logic Programs Research Articles

Justifications for programs with disjunctive and causal-choice rules

AbstractIn this paper, we study an extension of the stable model semantics for disjunctive logic programs where each true atom in a model is associated with an algebraic expression (in terms of rule labels) that represents its justifications. As in our previous work for non-disjunctive programs, these justifications are obtained in a purely semantic way, by algebraic operations (product, addition and application) on a lattice of causal values. Our new definition extends the concept ofcausal stable modelto disjunctive logic programs and satisfies that each (standard) stable model corresponds to a disjoint class of causal stable models sharing the same truth assignments, but possibly varying the obtained explanations. We provide a pair of illustrative examples showing the behaviour of the new semantics and discuss the need of introducing a new type of rule, which we callcausal-choice. This type of rule intuitively captures the idea of “Amay causeB” and, when causal information is disregarded, amounts to a usual choice rule under the standard stable model semantics.

A game semantics for disjunctive logic programming

Denotational semantics of logic programming and its extensions (by allowing negation, disjunctions, or both) have been studied thoroughly for many years. In 1998, a game semantics was given to definite logic programs by Di Cosmo, Loddo, and Nicolet, and a few years later it was extended to deal with negation by Rondogiannis and Wadge. Both approaches were proven equivalent to the traditional semantics. In this paper we define a game semantics for disjunctive logic programs and prove soundness and completeness with respect to the minimal model semantics of Minker. The overall development has been influenced by the games studied for PCF and functional programming in general, in the styles of Abramsky–Jagadeesan–Malacaria and Hyland–Ong–Nickau.

Semantics for Possibilistic Disjunctive Programs

AbstractIn this paper, a possibilistic disjunctive logic programming approach for modeling uncertain, incomplete, and inconsistent information is defined. This approach introduces the use of possibilistic disjunctive clauses, which are able to capture incomplete information and states of a knowledge base at the same time. By considering a possibilistic logic program as a possibilistic logic theory, a construction of a possibilistic logic programming semantic based on answer sets and the proof theory of possibilistic logic is defined. It shows that this possibilistic semantics for disjunctive logic programs can be characterized by a fixed-point operator. It is also shown that the suggested possibilistic semantics can be computed by a resolution algorithm and the consideration of optimal refutations from a possibilistic logic theory. In order to manage inconsistent possibilistic logic programs, a preference criterion between inconsistent possibilistic models is defined. In addition, the approach of cuts for restoring consistency of an inconsistent possibilistic knowledge base is adopted. The approach is illustrated in a medical scenario.

Comparison of Semantics of Disjunctive Logic Programs Based on Model-Equivalent Reduction

In this paper, it is shown that stable model semantics, perfect model semantics, and partial stable model semantics of disjunctive logic programs have the same expressive power with respect to the polynomial-time model-equivalent reduction. That is, taking perfect model semantics and stable model semantic as an example, any logic program P can be transformed in polynomial time to another logic program P′ such that perfect models (resp. stable models) of P 1-1 correspond to stable models (resp. perfect models) of P′, and the correspondence can be computed also in polynomial time. However, the minimal model semantics has weaker expressiveness than other mentioned semantics, otherwise, the polynomial hierarchy would collapse to NP.

Completeness of hyper-resolution via the semantics of disjunctive logic programs

We present a proof of completeness of hyper-resolution based on the fixpoint semantics of disjunctive logic programs. This shows that hyper-resolution can be studied from the point of view of logic programming.

Comparisons and computation of well-founded semantics for disjunctive logic programs

Much work has been done on extending the well-founded semantics to general disjunctive logic programs and various approaches have been proposed. However, these semantics are different from each other and no consensus is reached about which semantics is the most intended. In this article, we look at disjunctive well-founded reasoning from different angles. We show that there is an intuitive form of the well-founded reasoning in disjunctive logic programming which can be characterized by slightly modifying some existing approaches to defining disjunctive well-founded semantics, including program transformations, argumentation, unfounded sets (and resolution-like procedure). By employing the techniques developed by Brass and Dix in their transformation-based approach, we also provide a bottom-up procedure for this semantics. The significance of our work is not only in clarifying the relationship among different approaches, but also shed some light on what is an intended well-founded semantics for disjunctive logic programs.

Minimal founded semantics for disjunctive logic programs and deductive databases

In this paper, we propose a variant of stable model semantics for disjunctive logic programming and deductive databases. The semantics, called minimal founded, generalizes stable model semantics for normal (i.e. non-disjunctive) programs, but differs from disjunctive stable model semantics (the extension of stable model semantics for disjunctive programs). Compared with disjunctive stable model semantics, minimal founded semantics seems to be more intuitive, it gives meaning to programs which are meaningless under stable model semantics and is no harder to compute. More specifically, minimal founded semantics differs from stable model semantics only for disjunctive programs having constraint rules or rules working as constraints. We study the expressive power of the semantics, and show that for general disjunctive datalog programs it has the same power as disjunctive stable model semantics.

Collective Argumentation and Disjunctive Logic Programming

An extension of an abstract argumentation framework is introduced that provides a direct representation of global conflicts between sets of arguments. The extension, called collective argumentation, turns out to be suitable for representing semantics of disjunctive logic programs. Collective argumentation theories are shown to possess a four-valued semantics, and are closely related to multiple-conclusion (Scott) consequence relations. Two special kinds of collective argumentation, positive and negative argumentation, are considered in which the opponents can share their arguments. Negative argumentation turns out to be especially appropriate for analysing stable sets of arguments. Positive argumentation generalizes certain alternative semantics for logic programs.

Disjunctive logic programs with inheritance

The paper proposes a new knowledge representation language, called DLP<, which extends disjunctive logic programming (with strong negation) by inheritance. The addition of inheritance enhances the knowledge modeling features of the language providing a natural representation of default reasoning with exceptions. A declarative model-theoretic semantics of DLP< is provided, which is shown to generalize the Answer Set Semantics of disjunctive logic programs. The knowledge modeling features of the language are illustrated by encoding classical nonmonotonic problems in DLP<. The complexity of DLP< is analyzed, proving that inheritance does not cause any computational overhead, as reasoning in DLP< has exactly the same complexity as reasoning in disjunctive logic programming. This is confirmed by the existence of an efficient translation from DLP< to plain disjunctive logic programming. Using this translation, an advanced KR system supporting the DLP< language has been implemented on top of the DLV system and has subsequently been integrated into DLV.

Clausal Logic and Logic Programming in Algebraic Domains

We introduce a domain-theoretic foundation for disjunctive logic programming. This foundation is built on clausal logic, a representation of the Smyth powerdomain of any coherent algebraic dcpo. We establish the completeness of a resolution rule for inference in such a clausal logic; we introduce a natural declarative semantics and a fixed-point semantics for disjunctive logic programs, and prove their equivalence; finally, we apply our results to give both a syntax and semantics for default logic in any coherent algebraic dcpo.

Semantic Forcing in Disjunctive Logic Programs

We propose a semantics for disjunctive logic programs, based on the single notion of forcing. We show that the semantics properly extends, in a natural way, previous approaches. A fixpoint characterization is also provided. We also take a closer look at the relationship between disjunctive logic programs and disjunctive‐free logic programs. We present certain criteria under which a disjunctive program is semantically equivalent with its disjunctive‐free (shifted) version.

Semantics and expressiveness of disjunctive ordered logic

The paper proposes a knowledge representation language which extends logic programming with disjunction, inheritance, true negation and modularization. The resulting language is called Disjunctive Ordered Logic (\mathcal{DOL} for short). A modeldtheoretic semantics for \mathcal{DOL} is given, and it is shown to extend the stable model semantics of disjunctive logic programs. A number of examples show the suitability of \mathcal{DOL} for knowledge representation and commonsense reasoning. Among other things, the proposed language appears to be a powerful tool for the description of diagnostic processes which are based on stepwise refinements. The complexity and the expressiveness of the language are carefully analyzed. The analysis pays particular attention to the relative power and complexity of inheritance, negation and disjunction. An interesting result in this course concerns the role played by inheritance. Indeed, our results show that inheritance does not increase at all the complexity of any fragment of the language, while it does increase the expressive power of some \mathcal{DOL} fragments.

Argumentation in disjunctive logic programming

Argumentation (abduction) is widely applied in artificial intelligence (AI) and law reasoning. However, the problem of how to perform argumentation in disjunctive logic programming (DLP) is still open. In addition, a unifying semantic framework is required for incorporating various semantics for DLP. An argumentation-theoretic framework for DLP by taking the disjuncts of negative literals as abducibles is presented. This semantics not only is a simple and intuitive framework for performing argumentation and abduction in DLP, but also provides a unifying framework for many key semantics of disjunctive logic programs. In particular, it is shown that the EGCWA, well-founded model and disjunctive stable models can all be embedded into this semantics.

Expressiveness of stable model semantics for disjunctive logic programs with functions

In this paper, we study the expressive power and recursion-theoretic complexity of disjunctive logic programs with functions symbols over Herbrand models. In particular, we consider the disjunctive stable model semantics, and show that a relation R is definable over the Herbrand universe of a disjunctive logic program if and only if R is Π 1 1 definable. Thus, disjunctive logic programming under the stable model semantics expresses exactly Π 1 1, and is thus Π 1 1 complete over the integers. This result is surprising because it shows that disjunctive logic programming is not more expressive than normal logic programming under the stable or well-founded semantics. This sharply contrasts with the function-free case.

Disjunctive Stable Models: Unfounded Sets, Fixpoint Semantics, and Computation

Disjunctive logic programs have become a powerful tool in knowledge representation and commonsense reasoning. This paper focuses on stable model semantics, currently the most widely acknowledged semantics for disjunctive logic programs. After presenting a new notion of unfounded sets for disjunctive logic programs, we provide two declarative characterizations of stable models in terms of unfounded sets. One shows that the set of stable models coincides with the family of unfounded-free models (i.e., a model is stable iff it contains no unfounded atoms). The other proves that stable models can be defined equivalently by a property of their false literals, as a model is stable iff the set of its false literals coincides with its greatest unfounded set. We then generalize the well-founded WPoperator to disjunctive logic programs, give a fixpoint semantics for disjunctive stable models and present an algorithm for computing the stable models of function-free programs. The algorithm's soundness and completeness are proved and some complexity issues are discussed.

Disjunctive logic and semantics of disjunctive logic programs

In common sense reasoning with incomplete knowledge bases, conclusions are made by default. However, it is observed that when the negation-by-default operator not is defined as not provable, the disjunctive logic program {a νb, nota, notb} should be consistent becausea being not provable andb being not provable does not implya νb being not provable. Such an observation is significant for non-monotonic reasoning, but none of the major current semantics for disjunctive logic programs is able to support it because they are all based on classical first-order logic in which assuming nota and notb implies assuming not (a νb). A new first-order logic (disjunctive logic) is developed that fully complies with this observation and new semantics for disjunctive logic programs are established. This theory is able to formalize and solve some paradoxical, problems, such as the lottery paradox.

Combining Closed World Assumptions with Stable Negation

We study the semantics of disjunctive logic programs that simultaneously contain multiple kinds of default negations. We introduce operators not G, not W, and not STB in the language of logic programs to represent the Generalized Closed World Assumption, the Weak Generalized Closed World Assumption, and the stable negation, respectively. The notion of stratification involving different kinds of negations is defined and the meaning of stratified programs with multiple negations is described. The class of stratified programs is extended to the class of quasi-stratified programs and the semantics of the latest class is studied.

Autoepistemic logics as a unifying framework for the semantics of logic programs

In this paper, it is shown that a three-valued autoepistemic logic provides an elegant unifying framework for some of the major semantics of normal and disjunctive logic programs and logic programs with classical negation, namely, the stable semantics, the well-founded semantics, supported models, Fitting's semantics, Kunen's semantics, the stationary semantics, and answer sets. For the first time, so many semantics are embedded into one logic. The framework extends previous results—by Gelfond, Lifschitz, Marek, Subrahmanian, and Truszczynski —on the relationships between logic programming and Moore's autoepistemic logic. The framework suggests several new semantics for negation-as-failure. In particular, we will introduce the epistemic semantics for disjunctive logic programs. In order to motivate the epistemic semantics, an interesting class of applications called ignorance tests will be formalized; it will be proved that ignorance tests can be defined by means of the epistemic semantics, but not by means of the old semantics for disjunctive programs. The autoepistemic framework provides a formal foundation for an environment that integrates different forms of negation. The role of classical negation and various forms of negation-by-failure in logic programming will be briefly discussed.

Propositional semantics for disjunctive logic programs

In this paper we study the properties of the class of head-cycle-free extended disjunctive logic programs (HEDLPs), which includes, as a special case, all nondisjunctive extended logic programs. We show that any propositional HEDLP can be mapped in polynomial time into a propositional theory such that each model of the latter corresponds to an answer set, as defined by stable model semantics, of the former. Using this mapping, we show that many queries over HEDLPs can be determined by solving propositional satisfiability problems. Our mapping has several important implications: It establishes the NP-completeness of this class of disjunctive logic programs; it allows existing algorithms and tractable subsets for the satisfiability problem to be used in logic programming; it facilitates evaluation of the expressive power of disjunctive logic programs; and it leads to the discovery of useful similarities between stable model semantics and Clark's predicate completion.

An alternative approach to the semantics of disjunctive logic programs and deductive databases

In this paper, we study a new semantics of logic programming and deductive databases. Thepossible model semantics is introduced as a declarative semantics of disjunctive logic programs. The possible model semantics is an alternative theoretical framework to the classical minimal model semantics and provides a flexible inference mechanism for inferring negation in disjunctive logic programs. We also present a proof procedure for the possible model semantics and show that the possible model semantics has an advantage from the computational complexity point of view.